The VMC method provides a way to obtain accurate many-body wavefunctions for electronic systems and to statistically evaluate most observables using these wavefunctions. By computing all integral statistically, potentially any form of wavefunction is usable.
The variance of the local energy of the wavefunction is a measure of its accuracy. However, although the variational principle guarantees that energies are variational, no guarantees exist for other observables. The VMC method provides no direct insight into the parameterisation of a wavefunction, so that wavefunctions must be designed using physical insight. Hence although precise measurements of many-body observables is possible, there is only limited confidence in their accuracy for quantities other than the total energy.
The fixed-node DMC method is a near-exact method for finding the fermionic ground state of a Hamiltonian. The key approximation in the fixed-node DMC method is the use of an approximate nodal surface from a guiding wavefunction. In practice, the nodes of HF or DFT wavefunctions are very accurate. Good quality guiding wavefunctions are required to perform DMC calculations and optimised VMC wavefunction are sufficient.
DMC and VMC may be seen as complementing each other: DMC providing a very reliable estimate of the ground state energy, and the cheaper VMC potentially providing analytic many-body wavefunctions of similar accuracy.
The VMC and DMC methods are elegant, potentially very powerful methods for solving the Schrödinger equation. They must be judged by their performance in actual calculations on realistic systems: the methods must be sufficiently accurate and computationally affordable in systems meriting high accuracy investigation or in systems not readily accessed by other methods. Detailed descriptions of their implementation are described in the next chapter, and applications of the methods are presented in the following chapters.